\begin{center}{Abstract for {\bf Quantum generalized cohomology}}\bigskip
\end{center}
\noindent There is a variant of Segal's category of Riemann surfaces, in which
morphisms are stable complex algebraic curves [i.e. double points are
allowed], with some smooth points marked; composition is defined by
glueing at marked points. The spaces of morphisms in this category are
built from the compactified moduli spaces $\overline M_{g,n}$ of
Deligne, Mumford, and Knudesen; here $g$ is the genus and $n$ is the number
of marked points. A generalized topological
field theory taking values in the category of module-spectra over a
ring-spectrum $\bf R$ is a family $$\tau_{g,n} : \overline M_{g,n} \rightarrow
{\bf M} \wedge_{\bf R} \dots \wedge_{\bf R} {\bf M} = {\bf M}^{\wedge n}$$ of
maps, which
respect composition of morphisms. More precisely, $\bf M$ is an $\bf R$-module
spectrum, $\wedge_{\bf R}$ is the Robinson smash product, and $\bf M$ is
endowed with a suitably nondegenerate bilinear form $${\bf M} \wedge_{\bf R}
{\bf M} \rightarrow {\bf R}.$$ This data entails the existence of
an $\bf R$-algebra structure on $\bf M$, such that $\tau_{g,1}$ is a morphism
of monoids if the moduli space of curves is given the pair-of-pants
product; it seems to define a natural context for quantum
generalized cohomology.\medskip
\noindent There is an interesting example of all this, associated to a smooth algebraic
variety $V$. It is closely related to the Tate $\bf MU$-cohomology of the
universal cover of the free loopspace of $V$, but it can be described more
concretely in terms of the rational Novikov ring $\Lambda = {\Bbb Q}
[H_{2}(V,{\Bbb Z})]$ of $V$ by setting ${\bf R} = {\bf MU} \otimes
\Lambda$; then {\bf E} is the function spectrum $F(V,{\bf R})$
representing the cobordism of $V$ tensored with $\Lambda$, and the
bilinear pairing is defined by Poincar\'e duality. In this case $\tau_{g,n}$
represents the cobordism class of the space of stable maps
[in the sense of Kontsevich] from a curve of genus $g$, marked with $n$
ordered smooth points together with an indeterminate number of unordered
smooth points, to $V$. A variant construction requires the unordered points
to lie on a cycle $z$ in $V$; this defines a parameterized family of
multiplications satisfying the analogue of the WDVV equation. When $V$ is a point, the
resulting theory boils down to the version of topological gravity I
advertised at the Adams Symposium; the coupling constant of the associated topological
field theory is the cobordism analogue of Manin's exponential
$$\sum_{n \geq 0} \overline M_{0,n+3} \frac {z^{n}}{n!} .$$ Although
much of the machinery used here comes from fields
adjacent to topology, this paper is concerned with the old problem of
constructing complex cobordism out of Riemann surfaces by some
analogue of the plus-construction. Having hacked through the physics
background, I hope to produce a more topological account in the near future.
\medskip
\noindent This is to appear in Contemporary Math., in the Proceedings of the
Hartford/Luminy Conference on the Renaissance of Operads, ed. J.-L.
Loday, J. Stasheff, and A. A. Voronov.
\end{document}
[ Jack tells me that this is the final version. I've labeled the
DVI file as Luniny6-final.dvi and reclused the original version, CWW
7/13/98]